Cuba is annexed in '98?
I think statehood could probably be brought about relatively quickly. By 1920 Cuba is a state. The Easter Uprising in Ireland could have a salutary effect on American political thinking, as politicians in Washington start thinking it would be a good idea to get Cuba all the way inside the tent before embarking on any foreign adventures. Furthermore, the dominance in Cuba of a political machine that aligns itself with the Democratic Party means that Mr. Wilson has electoral reasons to bring Cuba into the Union.
Being a United States territory and then state is really a net gain for the Cubans. Cuba's economy depended mainly on the US anyway before the US cut them off with sanctions. With statehood they have control of their own internal affairs, free-trade access to their biggest trading partner, and some influence on that trading partner's affairs. Cuba is a beneficiary of the New Deal, and of federal law enforcement. It has some level of democratic participation, since it is an American state, so though a corrupt oligarchy may rule, the kind of dictatorship that was seen in OTL can never arise.
I don't think you'd see a Jim Crow-style regime in Cuba, since there was little base for it in Cuba's history. Cuba's legal system is probably going to stick with some kind of adapted Spanish system, similar to how Louisiana has Napoleonic Code-based system. After all, though the United States may have conquered it, the population will still have the centuries long legacy of Spanish rule.